On this page you’ll learn an introduction to **two essential trigonometric functions**, the **sine and cosine functions**. These functions relate the angles and sides of a right triangle and have numerous applications in fields such as engineering, physics, and astronomy. The page includes the definitions, formulas, and tables of values for these functions, making it a valuable resource for anyone studying or working in technical fields.

## Sine Theta (sin Î¸)

The **sine function** is a trigonometric function that relates the angle Î¸ to the ratio of the length of the side opposite to the angle to the length of the hypotenuse of a right triangle.

**Formula:**sin Î¸ = opposite/hypotenuse**Unit circle:**The sine function can also be represented on the unit circle, which is a circle with a radius of 1 unit, where the angle Î¸ is measured from the positive x-axis in the counterclockwise direction. The sine function of an angle Î¸ is equal to the y-coordinate of the point on the unit circle that corresponds to the angle Î¸.**Periodicity:**The sine function is periodic, with a period of 2Ï€. This means that sin(Î¸ + 2Ï€) = sin(Î¸) for any angle Î¸.**Graph:**The graph of the sine function is a periodic wave that oscillates between -1 and 1, with a wavelength of 2Ï€.

Î¸ | 0Â° | 30Â° | 45Â° | 60Â° | 90Â° | 180Â° | 270Â° | 360Â° |
---|---|---|---|---|---|---|---|---|

sin Î¸ | 0 | 1/2 | âˆš2/2 | âˆš3/2 | 1 | 0 | -1 | 0 |

## Cosine Theta (cos Î¸)

The **cosine function** is a trigonometric function that relates the angle Î¸ to the ratio of the length of the side adjacent to the angle to the length of the hypotenuse of a right triangle.

**Formula**: cos Î¸ = adjacent/hypotenuse**Unit circle:**The cosine function can also be represented on the unit circle. The cosine function of an angle Î¸ is equal to the x-coordinate of the point on the unit circle that corresponds to the angle Î¸.**Periodicity:**The cosine function is periodic, with a period of 2Ï€. This means that cos(Î¸ + 2Ï€) = cos(Î¸) for any angle Î¸.**Graph:**The graph of the cosine function is also a periodic wave that oscillates between -1 and 1, with a wavelength of 2Ï€.

Î¸ | 0Â° | 30Â° | 45Â° | 60Â° | 90Â° | 180Â° | 270Â° | 360Â° |
---|---|---|---|---|---|---|---|---|

cos Î¸ | 1 | âˆš3/2 | âˆš2/2 | 1/2 | 0 | -1 | 0 | 1 |

These tables can be useful when solving problems that involve trigonometric functions, such as finding the length of a side of a right triangle or the value of an angle given the lengths of two sides.